Optimal. Leaf size=200 \[ \frac{\tan (c+d x) \left (4 a^2 b B+a^3 (-C)+4 a b^2 (3 A+2 C)+4 b^3 B\right )}{6 b d}+\frac{\left (4 a^2 (2 A+C)+8 a b B+b^2 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec (c+d x) \left (-2 a^2 C+8 a b B+12 A b^2+9 b^2 C\right )}{24 d}+\frac{(4 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^2}{12 b d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^3}{4 b d} \]
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Rubi [A] time = 0.348008, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.18, Rules used = {4082, 4002, 3997, 3787, 3770, 3767, 8} \[ \frac{\tan (c+d x) \left (4 a^2 b B+a^3 (-C)+4 a b^2 (3 A+2 C)+4 b^3 B\right )}{6 b d}+\frac{\left (4 a^2 (2 A+C)+8 a b B+b^2 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec (c+d x) \left (-2 a^2 C+8 a b B+12 A b^2+9 b^2 C\right )}{24 d}+\frac{(4 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^2}{12 b d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^3}{4 b d} \]
Antiderivative was successfully verified.
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Rule 4082
Rule 4002
Rule 3997
Rule 3787
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sec (c+d x) (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x))^2 (b (4 A+3 C)+(4 b B-a C) \sec (c+d x)) \, dx}{4 b}\\ &=\frac{(4 b B-a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 b d}+\frac{C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x)) \left (b (12 a A+8 b B+7 a C)+\left (12 A b^2+8 a b B-2 a^2 C+9 b^2 C\right ) \sec (c+d x)\right ) \, dx}{12 b}\\ &=\frac{\left (12 A b^2+8 a b B-2 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac{(4 b B-a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 b d}+\frac{C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 b d}+\frac{\int \sec (c+d x) \left (3 b \left (8 a b B+4 a^2 (2 A+C)+b^2 (4 A+3 C)\right )+4 \left (4 a^2 b B+4 b^3 B-a^3 C+4 a b^2 (3 A+2 C)\right ) \sec (c+d x)\right ) \, dx}{24 b}\\ &=\frac{\left (12 A b^2+8 a b B-2 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac{(4 b B-a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 b d}+\frac{C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 b d}+\frac{\left (4 a^2 b B+4 b^3 B-a^3 C+4 a b^2 (3 A+2 C)\right ) \int \sec ^2(c+d x) \, dx}{6 b}+\frac{1}{8} \left (8 a b B+4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) \int \sec (c+d x) \, dx\\ &=\frac{\left (8 a b B+4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left (12 A b^2+8 a b B-2 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac{(4 b B-a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 b d}+\frac{C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 b d}-\frac{\left (4 a^2 b B+4 b^3 B-a^3 C+4 a b^2 (3 A+2 C)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 b d}\\ &=\frac{\left (8 a b B+4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left (4 a^2 b B+4 b^3 B-a^3 C+4 a b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 b d}+\frac{\left (12 A b^2+8 a b B-2 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac{(4 b B-a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 b d}+\frac{C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 b d}\\ \end{align*}
Mathematica [A] time = 1.55599, size = 300, normalized size = 1.5 \[ -\frac{\sec ^4(c+d x) \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right ) \left (12 \cos ^4(c+d x) \left (4 a^2 (2 A+C)+8 a b B+b^2 (4 A+3 C)\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-2 \sin (c+d x) \left (4 \cos (c+d x) \left (9 a^2 B+2 a b (9 A+10 C)+10 b^2 B\right )+3 \cos (2 (c+d x)) \left (4 a^2 C+8 a b B+4 A b^2+3 b^2 C\right )+12 a^2 B \cos (3 (c+d x))+12 a^2 C+24 a A b \cos (3 (c+d x))+24 a b B+16 a b C \cos (3 (c+d x))+12 A b^2+8 b^2 B \cos (3 (c+d x))+21 b^2 C\right )\right )}{48 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 321, normalized size = 1.6 \begin{align*}{\frac{{a}^{2}A\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{B{a}^{2}\tan \left ( dx+c \right ) }{d}}+{\frac{{a}^{2}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{2}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+2\,{\frac{Aab\tan \left ( dx+c \right ) }{d}}+{\frac{Bab\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{d}}+{\frac{Bab\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{4\,abC\tan \left ( dx+c \right ) }{3\,d}}+{\frac{2\,abC\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{A{b}^{2}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{A{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{2\,B{b}^{2}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{B{b}^{2}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{{b}^{2}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{3\,{b}^{2}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{b}^{2}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02902, size = 413, normalized size = 2.06 \begin{align*} \frac{32 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a b + 16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B b^{2} - 3 \, C b^{2}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a^{2}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 24 \, B a b{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, A b^{2}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, A a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 48 \, B a^{2} \tan \left (d x + c\right ) + 96 \, A a b \tan \left (d x + c\right )}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.559991, size = 510, normalized size = 2.55 \begin{align*} \frac{3 \,{\left (4 \,{\left (2 \, A + C\right )} a^{2} + 8 \, B a b +{\left (4 \, A + 3 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (4 \,{\left (2 \, A + C\right )} a^{2} + 8 \, B a b +{\left (4 \, A + 3 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (8 \,{\left (3 \, B a^{2} + 2 \,{\left (3 \, A + 2 \, C\right )} a b + 2 \, B b^{2}\right )} \cos \left (d x + c\right )^{3} + 6 \, C b^{2} + 3 \,{\left (4 \, C a^{2} + 8 \, B a b +{\left (4 \, A + 3 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{2} + 8 \,{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sec{\left (c + d x \right )}\right )^{2} \left (A + B \sec{\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21748, size = 851, normalized size = 4.26 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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